On Z p -extensions of real abelian number fields

Transcription

1 Università degli Studi di Roma La Sapienza Facoltà di Scienze Matematiche Fisiche e Naturali Dottorato di Ricerca in Matematica XX Ciclo On Z p -extensions of real abelian number fields Candidato Filippo A. E. Nuccio Mortarino Majno di Capriglio Relatore prof. René Schoof Commissione prof. Massimo Bertolini prof. Roberto Dvornicich prof. Riccardo Salvati Manni

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3 Que otros se enorgullezcan por lo que han escrito, yo me enorgullezco por lo que he leído. Jorge Luis Borges, Elogio de la sombra 1969

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5 Acknowledgments There are many people I shall need to thank for the help I received in these four years - both inside and outside mathematical departments. Nevertheless, I want to thank three persons in particular because I owe them most of what I have learnt during my PhD and because they have helped me in so many different ways. The first one is my advisor, René Schoof. His guidance and help have been extremely reassuring and I am unable to list all the Mathematics I have learnt from him. Let me just mention the two class field theory courses he gave during my first two years of PhD and the peculiar point of view he shared with me about Iwasawa theory. Seeing him at work has surely been the best stimulus for studying algebraic geometry. Secondly, I would like to thank Daniel Barsky for proposing me to work on p-adic zeta functions and for the groupe de travail we organised while I was in Paris in Spring Although I was unable to pursue the project we undertook together, I found his help tremendously influencing to understand the p-adic analysis involved in Iwasawa theory. Thirdly, I cannot forget that almost all the Iwasawa theory I know and most of the cohomological techniques I can use derive from the afternoons I spent in the office of David Vauclair, both during my stay in Caen in 2007 and afterwards. I finally wish to thank Gabriel Chênevert for a careful reading of this thesis and for the discussions we had both in Paris and in Leiden. Despite their informal character, I hardly realize how instructive they were. Con scadenza grosso modo settimanale, il sito internet di Repubblica propone un sondaggio sull attualità politica e sociale. Una percentuale oscillante fra l 1% e il 3% di coloro che esprimono un opinione si reca volontariamente e spontaneamente sulla pagina del sondaggio per votare Non so. Non essendo in grado di apprezzare pienamente il significato del gesto, intendo dedicare loro questa tesi. v

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7 Contents Introduction 9 A Criterion for Greenberg s Conjecture 17 Cyclotomic Units and Class Groups in Z p -extensions of Real Abelian Number Fields 23 On Fake Z p -extensions of Number Fields 43 vii

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9 Introduction Let K be a number field and let p be a prime number. It is by now a very classical result that the growth of the p-part of the class number along any Z p -extension K /K is controlled by an asymptotical formula. More precisely, we have the following Theorem 1 (Iwasawa, [Iwa73]). Let K /K be a Z p -extension and p en be the order of the p-sylow subgroup of the class group of K n, the subfield of degree p n. Then there exist three integers µ, λ, ν such that e n = µp n + λn + ν for all n 0. In their celebrated work [FW79] Bruce Ferrero and Lawrence Washington proved that µ = 0 if K /K is the cyclotomic Z p -extension of an abelian base field. In 1976 Ralph Greenberg studied in his thesis [Gre76] some criterion for λ to be 0 when the ground field is totally real and the extension is the cyclotomic one. Since then, the condition λ = 0 has become a conjecture, known as Greenberg s Conjecture although Greenberg himself never stated it as such. The conjecture has been verified computationally in many cases using different techniques: see, for instance, [FT95], [IS97], [KS95], [Nis06], [OT97], [Tay00] and the references there. Greenberg s conjecture may be seen as a generalization of a long-standing conjecture by Vandiver, predicting that for every prime number p, the class number of Q(ζ p ) +, the totally real subfield of the p-th cyclotomic field, is never divisible by p. Indeed, we have the following well-known result: Theorem 2 ([Was97], Proposition 13.36). Let K be a number field in which there is a unique prime above the prime number p and let K /K be a totally ramified Z p -extension of K. Then p Cl K p Cl Kn for all n 0. 9

10 Since Q(ζ p ) + /Q(ζ p ) + satisfies the hypothesis of the theorem, Vandiver s conjecture would clearly imply λ = 0 for this extension. On the other hand, it is still unknown whether λ = 0 for the extension would imply that p Cl Q(ζp) +. Observe that Vandiver s conjecture has been checked numerically for all p < in [BCE + 01] but for the moment the best theoretical result on the conjecture is the following Theorem 3 (Soulé, [Sou99]). Let i be odd and assume p > i 224i4. Then the eigenspace of Cl Q(ζp) Z p on which Gal(Q(ζ p )/Q) acts as ω p i is trivial, where ω is the Teichmüller character. We remark that in Washington s book [Was97] a heuristic argument is presented (see Chapter 8, 3), according to which Vandiver s Conjecture should be false. The argument is based on the idea that the probability that an eigenspace is non-trivial is equally distributed in the interval 1 i p (for odd i s): it might thus need some refinement in view of Soulé s result above. On the other side, there are some theoretical argument that might suggest the validity of Greenberg s Conjecture. In 1995, James Kraft and René Schoof proposed in [KS95] a procedure to check Greenberg s Conjecture - the paper only deals with the case of a real quadratic number field in which p does not split, but this plays a minor role in their argument - and their work gives strong theoretical evidence for the conjecture. The idea is to work with cyclotomic units rather then with ideal classes: indeed, a celebrated theorem by Warren Sinnott (see [Sin81]) shows that for all n 0 the class number of K n (assuming that the degree [K : Q] is prime to p) coincides up to factors prime to p with the index of the submodule of cyclotomic units inside the full group of units O K n. One can therefore check if the class numbers stabilize by checking the stabilization of the index of these cyclotomic units. Call B n the quotient of O K n by the cyclotomic units at level n: by a very elegant, but elementary, commutative algebra argument over the ring R n := Z/p n+1 Z[G n ] (where G n = Gal(K n /K)), Kraft and Schoof can describe concretely the structure of Hom(B n, Q p /Z p ) - itself again of the same order as B n and Cl Kn (up to p-units). They show it is cyclic over R n and the ideal of relations is generated by Frobenius elements of primes l 1,, l k that split completely in K n (ζ p n+1). If one can prove that at least two of these Frobenius elements are prime to each other, then the ideal of relations is the whole ring, the module is trivial and the conjecture is verified. This is not always the case, but the same strategy shows that if for every n there exists two Frobenius elements that generate an ideal in R n whose index is independent of n, then the conjecture holds 10

11 true. Looking at the computations gathered in the paper, as n grows and l runs through many totally split primes, the elements of R n corresponding to Frob l look random, and one can therefore expect the conjecture to hold. Another reason to believe Greenberg s Conjecture may come from the so-called Main Conjecture (now a theorem, proven by Barry Mazur and Andrew Wiles, [MW84]). Let K be a totally real number field, that we also assume to be abelian, and let F = K(ζ p ): F is then a CM field and we denote by F + its maximal real subfield. Consider the cyclotomic Z p - extension F /F (and analogously F /F + + ) and set Γ = Gal(F /F ) = Gal(F /F + + ), = Gal(F/K) and + = Gal(F + /K). We also put G = Gal(F /K) = Γ and G + = Gal(F /K) + = Γ + and accordingly define their Iwasawa algebras Λ(G) and Λ(G + ) where, for any profinite group Π, we set Λ(Π) = lim Z p [Π/H] for H running through all open, normal subgroups of Π. We therefore get a diagram of fields F F + Γ G + Γ c F + + F G K Denoting by L n the p-hilbert class field of F n, the extension L /F is the maximal everywhere unramified abelian p-extension of F where we have set L = L n. The Galois group X := Gal(L /F ) is a finitely generated Z p -module (see [FW79]) and it carries a natural action of G coming from the Artin isomorphism X = lim (Cl Fn Z p ), the projective limit being taken with respect to norm maps: it is therefore a Λ(G)-module. 11

12 The unique element c of order 2 in acts semisimply on every Λ(G)- module M and decomposes it canonically as M = M + M (1) where c acts trivially on M + and as 1 on M. We can apply this both to Λ(G) itself, finding that Λ(G) + = Λ(G + ) (see [CS06], Lemma 4.2.1) and to X: it is then a standard fact that X + = Gal(L + /F ) + as Λ(G + )-modules, where L + L is the maximal everywhere unramified abelian p-extension of F. + By the classical theory of Iwasawa algebras (see the Appendix of [CS06]) there exists characteristic ideals, say I Λ(G) and I + Λ(G + ) such that Λ(G)/I X and Λ(G + )/I + X + where denotes pseudoisomorphism; moreover, the Iwasawa algebras Λ(G) and Λ(G + ) are each isomorphic, as Λ(Γ)-modules, to respectively and /2 = + copies of Z p [[T ]], itself isomorphic to Z p [[Γ]] by sending a topological generator γ of Γ to 1 + T. Through these isomorphisms the ideals I and I + become generated by suitable collections of polynomials, say I = f 1,..., f and I + = f 1 +,..., f + /2. One checks easily that these polynomials are related to the Iwasawa invariants by λ(x) = deg(f i ) and λ(x + ) = deg(f i + ). Therefore Greenberg s conjecture says that all polynomials f i + (T ) are constant or, equivalently (the equivalence follows from [FW79]), that I + = Λ(G + ). Before passing to the analytic side of the Main Conjecture, we remark that our notation I + is consistent with (1), because the characteristic ideal of X + and the +-part of the characteristic ideal of X, seen as a Λ(G)-module, coincide. On the analytic side, Iwasawa together with Kubota and Leopoldt proved the existence of a p-adic pseudo-measure (see [Ser78]) ζ K,p such that 1 χ n K,pdζ K,p = E(p)ζ K (1 n) for all integers n 1 (2) G where ζ K (s) is the usual complex Dedekind zeta-function of K, χ K,p is the p-adic cyclotomic character of Gal( K/K) and E(p) is an Euler factor that is never 0 (mod p). By a classical theorem in p-adic analysis due to Kurt Mahler (see [Mah58]), there exists a correspondence M - called the Mahler transform - between the p-adic measures on G (resp. on G + ) and the Iwasawa algebra Λ(G) (resp. Λ(G + )). Let Θ(G) and Θ(G + ) be the 1 We recall that a p-adic measure µ on G is a continuous functional µ : C(G, C p) C p subject to the condition µ(f) Z p when f(g) Z p. A pseudo-measure ζ is an element of the total quotient ring of the Z p-algebra of p-adic measures such that ζ(1 g) is a measure for all g G. 12

13 augmentation ideals in Λ(G) and Λ(G + ), respectively: since ζ K,p is a pseudomeasure we have M(ζ K,p )Θ(G) Λ(G). Moreover, since the Dedekind zeta function ζ K vanishes for all even negative integers, (2) shows that all odd powers of the cyclotomic character have trivial integral against ζ K,p ; carefully pinning down the -action one sees that this forces ζ K,p to be in the +-part and we have M(ζ K,p )Θ(G + ) Λ(G + ). We need a final remark: an old theorem of Iwasawa (see [Iwa73]) shows that there exists a finitely generated, torsion Λ(G)-module X such that X + X. We do not discuss this module, and simply use this result to find an action of Λ(G + ) on I. Then the Main Conjecture states that I = M(ζ K,p )Θ(G + ) (3) as ideals of Λ(G + ). We want now to show that on one hand Greenberg s Conjecture implies Mazur and Wiles result, and on the other hand that the conjecture would follow from a more general Main Conjecture, namely I = M(ζ K,p )Θ(G);. (4) as Λ(G)-ideals. To do this, we recall the following fundamental theorem of Iwasawa: Theorem 4 (Iwasawa, [Iwa64]). Assume that for every n there is only one prime p n above p in F n. Let U 1 n be the local units of F + n,p n that are 1 (mod p n ) and let C 1 n be the p-adic completion of the image of cyclotomic units in U 1 n. Let U 1 and C 1 be the projective limits with respect to norms: then the characteristic ideal of U 1 /C 1 as Λ(G + )-module is M(ζ K,p )Θ(G + ). The assumption that there is a unique prime in F n above p is clearly not necessary, but it simplifies drastically our exposition. Class field theory gives an exact sequence 0 E 1 /C 1 U 1 /C 1 X + X + 0, where E 1 is the projective limit of global units of F + n that are 1 (mod p n ). Since the characteristic ideal of the second module is described by Theorem 4 and that of the third module is I by Iwasawa s result quoted above, X + = 0 implies a divisibility I M(ζ K,p )Θ(G + ): then the classical Class Number Formula turns this divisibility into an equality, giving (3). On the other hand, assume (4): writing X = X + X we get I = I + I and I + I = ( M(ζ K,p )Θ(G) ) + ( M(ζK,p )Θ(G) ) 13

14 as Λ(G)-modules. But since ζ K,p Λ(G) +, we find that ( M(ζ K,p )Θ(G) ) = 0 and hence I + I = ( M(ζ K,p )Θ(G) ) + = M(ζK,p )Θ(G + ) as Λ(G + )-modules: combining this with (3) we get I + = Λ(G + ), which is precisely Greenberg s Conjecture. The first two chapters of my thesis deal with Greenberg s conjecture. The first reproduces the work [CN08a] - written with Luca Caputo - and presents a condition for λ = 0 for some abelian field. This condition is far from being necessary. The second chapter is the paper [Nuc09] and it investigates a consequence of the conjecture for abelian number fields in which the rational prime p splits completely: namely, it shows that the equality of orders B n = Cl Kn Z p coming from Sinnott s work hinted at above, does not imply that a certain natural map between these groups is an isomorphism, and explicitely computes the kernel and the cokernel of the map. The interest of this analysis comes again from [KS95], where it was shown that if p does not split in K, then Greenberg s conjecture implies that the above natural map is indeed an isomorphism. The third chapter is the work [CN08b], again written with Luca Caputo, and deals with a non-galois extension in Iwasawa Theory. This is the definition that we propose: Definition 1. Let p be a prime number, let K be a number field and let K /K be a non-galois extension. Suppose that there exists a Galois extension L/K disjoint from K /K such that LK is a Galois closure of K /K. If LK /L is a Z p -extension, then K /K is called a fake Z p -extension. We then prove that the same Iwasawa formula as in Theorem 1 above holds also for the fake Z p -extension where K = Q, L is imaginary quadratic and L = LK is the anti-cyclotomic Z p -extension of L, so that K is the subextension of L fixed by any subgroup of order 2 inside the pro-dihedral group Gal(L /Q). In the last section of the paper we also investigate the algebraic structure of the projective limit of the class groups along this fake Z p -extension. References [BCE + 01] Joe Buhler, Richard Crandall, Reijo Ernvall, Tauno Metsänkylä, and M. Amin Shokrollahi, Irregular primes and cyclotomic invariants to 12 million, J. Symbolic Comput. 31 (2001), no. 1-2, 14

15 89 96, Computational algebra and number theory (Milwaukee, WI, 1996). [CN08a] [CN08b] [CS06] [FT95] [FW79] [Gre76] [IS97] [Iwa64] [Iwa73] [KS95] [Mah58] Luca Caputo and Filippo Alberto Edoardo Nuccio, A criterion for Greenberg s conjecture, Proc. Amer. Math. Soc. 136 (2008), no. 8, , On fake Z p -extensions of number fields, submitted, arxiv: 0807:1135. J. Coates and R. Sujatha, Cyclotomic fields and zeta values, Springer Monographs in Mathematics, Springer-Verlag, Berlin, Takashi Fukuda and Hisao Taya, The Iwasawa λ-invariants of Z p -extensions of real quadratic fields, Acta Arith. 69 (1995), no. 3, Bruce Ferrero and Lawrence C. Washington, The Iwasawa invariant µ p vanishes for abelian number fields, Ann. of Math. (2) 109 (1979), no. 2, Ralph Greenberg, On the Iwasawa invariants of totally real number fields, Amer. J. Math. 98 (1976), no. 1, Humio Ichimura and Hiroki Sumida, On the Iwasawa invariants of certain real abelian fields, Tohoku Math. J. (2) 49 (1997), no. 2, Kenkichi Iwasawa, On some modules in the theory of cyclotomic fields, J. Math. Soc. Japan 16 (1964), , On Z l -extensions of algebraic number fields, Ann. of Math. (2) 98 (1973), James S. Kraft and René Schoof, Computing Iwasawa modules of real quadratic number fields, Compositio Math. 97 (1995), no. 1-2, , Special issue in honour of Frans Oort. K. Mahler, An interpolation series for continuous functions of a p-adic variable, J. Reine Angew. Math. 199 (1958), [MW84] B. Mazur and A. Wiles, Class fields of abelian extensions of Q, Invent. Math. 76 (1984), no. 2,

16 [Nis06] [Nuc09] [OT97] [Ser78] [Sin81] [Sou99] [Tay00] [Was97] Yoshinori Nishino, On the Iwasawa invariants of the cyclotomic Z 2 -extensions of certain real quadratic fields, Tokyo J. Math. 29 (2006), no. 1, Filippo Alberto Edoardo Nuccio, Cyclotomic units and class groups in Z p -extensions of real abelian fields, Math. Proc. Cambridge Philos. Soc. (2009), to appear, arxiv: 0821:0784. Manabu Ozaki and Hisao Taya, On the Iwasawa λ 2 -invariants of certain families of real quadratic fields, Manuscripta Math. 94 (1997), no. 4, Jean-Pierre Serre, Sur le résidu de la fonction zêta p-adique d un corps de nombres, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 4, A183 A188. W. Sinnott, On the Stickelberger ideal and the circular units of an abelian field, Invent. Math. 62 (1980/81), no. 2, C. Soulé, Perfect forms and the Vandiver conjecture, J. Reine Angew. Math. 517 (1999), Hisao Taya, Iwasawa invariants and class numbers of quadratic fields for the prime 3, Proc. Amer. Math. Soc. 128 (2000), no. 5, Lawrence C. Washington, Introduction to cyclotomic fields, second ed., Graduate Texts in Mathematics, vol. 83, Springer- Verlag, New York,

17 A Criterion for Greenberg s Conjecture Luca Caputo and Filippo Alberto Edoardo Nuccio June 8 th, 2007 Abstract We give a criterion for the vanishing of the Iwasawa λ invariants of totally real number fields K based on the class number of K(ζ p ) by evaluating the p-adic L functions at s = Mathematical Subject Classification: Primary 11R23; Secondary 11R70 1 Introduction Let K be a real abelian number field and let p be an odd prime. Set F = K(ζ p ) where ζ p is a primitive p-th rooth of unity and H = Gal(F/K). Set, moreover, G = Gal(F/Q) and ϖ = Gal(Q(ζ p )/Q). So the diagram of our extensions is as follows: F = K(ζ p ) H K G Q(ζ p ) ϖ Q Let ω : H Z p and ω : ϖ Z p be the Teichmüller characters of K and Q, respectively. We give (Theorem 2.3) a criterion under which a set of odd Iwasawa invariants associated to F vanish: by means of a Spiegelungssatz, these odd invariants make their even mirrors vanish too. In the case p = 3 17

18 (Corollary 2.5) or p = 5 and [K : Q] = 2 (Theorem 2.7) this allows us to verify a conjecture of Greenberg for the fields satisfying our criterion. 2 Main result Proposition 2.1. The following equality holds rk p ( K2 (O K ) ) = rk p ( (Cl F ) ω 1) + S where K 2 (O K ) is the tame kernel of K-theory, Cl F is the class group of the ring O F [1/p] (and we take its ω 1 -component for the action of H) and S is the set of p-adic primes of K which split completely in F. Proof. This result dates back to Tate: for an explicit reference see [Gra] Theorem Proposition 2.2. Suppose that Q(ζ p ) is linearly disjoint from K over Q. Then the following equalities holds v p ( K2 (O K ) ) = v p ( ζk ( 1) ) if p 5 v 3 ( K2 (O K ) ) = v 3 ( ζk ( 1) ) + 1 where v p denotes the standard p-adic valuation and ζ K is the Dedekind zeta function for K. Proof. The Birch-Tate conjecture which has been proved by Mazur, Wiles and by Greither (since it is a consequence of the Main Conjecture in Iwasawa theory) tells that K 2 (O K ) = ζ K ( 1) w 2 where w 2 = max{n N the exponent of Gal(K(ζ n )/K) is 2} By our hypothesis, Q(ζ p ) is linearly disjoint from K over Q. Hence F/K is Galois with cyclic Galois group of order p 1. If p = 3, then for the same argument 3 w 2 but 9 w 2 since K(ζ 9 )/K has degree 6. Taking p-adic valuation we get the claim. Theorem 2.3. Let p 5. Suppose that the following holds K and Q(ζ p ) are linearly disjoint over Q; 18

19 the set S of Proposition (2.2) is empty; the Main Conjecture of Iwasawa theory holds for F. Then, if p does not divide the order of Cl F ( ω 1 ), λ χω 2(F ) = 0 for all characters χ of. Proof. First of all, we should just prove the theorem for non-trivial characters of, since λ ω 2 = 0 as it corresponds to the ω 2 -part of the cyclotomic extension of Q(ζ p ), which is always trivial: indeed, B 1/2 = 1/2, and then Herbrand s theorem and Leopoldt s Spiegelungssatz ([Was], theorems 6.7 and 10.9) give λ ω 2 = 0. By hypothesis, the set S of Proposition (2.1) is empty. Therefore rk p (K 2 (O K )) = 0 and Proposition (2.2) (that we can apply because K verifies its hypothesis) together with p 5 tells us that v p (ζ K ( 1)) = 0. Since we can factor ζ K (s) = L(s, χ) we find that (χ ˆ ) L(s, χ) = ζ Q (s) χ 1 v p ( ζk ( 1) ) = χ 1 v p ( L( 1, χ) ) = 0 (2.1) The interpolation formula for the p-adic L-function (see [Was], chapter 5) tells us that L p ( 1, χ) = ( 1 χω 2 (p)p ) L( 1, χω 2 ); (2.2) now we invoke the Main Conjecture as stated in ([Gre], page 452) to relate these L functions with the characteristic polynomials of some sub-modules of the Iwasawa module X (F ). Observe that the hypothesis of linear disjointness tells us that Ĝ = ˆ ˆϖ so we can split X (F ) = χ p 1 X (F )(χω i ) i=1 where G acts on X (F )(χω i ) as g x = (χω i )(g)x for all g G and x X. Then the Main Conjecture for F allows us to write L p ( 1, χω i ) = f( p/(1 + p), χ 1 ω 1 i ) for all even 2 i p 1, where f(t, χ 1 ω 1 i ) Z p [T ] is the characteristic polynomial of X (F )(χ 1 ω 1 i ): thus L p ( 1, χω i ) is ( Z p -integral. ) Applying this for i = 2 and plugging it in (2.2) ( we find ) v p L( 1, χ) 0 for all χ, and thanks to (2.1) we indeed find vp L( 1, χ) = 0 for all χ ˆ, so v p ( Lp ( 1, χω 2 ) ) = 0 χ ˆ. 19

20 If we now apply again the Main Conjecture we find that this corresponds to v p ( f ( p 1, χ 1 ω 1)) = v p ( f ( p 1 + p, χ 1 ω 1)) = 0 χ ˆ. Since f(t, χ 1 ω 1 ) Z p [T ], ( is distinguished (see [Was], chapter 7) this is possible if and only if deg T f(t, χ 1 ω 1 ) ) = 0; but this is precisely the Iwasawa invariant λ χ 1 ω 1, so we have λ χ 1 ω 1 = 0 χ ˆ. Since the inequality λ χ 1 ω 1 λ χω2 is classical and well-known (see, for instance, [BN] section 4), we achieve the proof. Remark 2.4. We should ask that the Main Conjecture holds for K to apply it in the form of [Gre]. For this, it is enough that there exists a field E that is unramified at p and such that F = E(ζ p ), as it is often the case in the applications. Moreover, we remark that the hypotheses of the theorem are trivially fulfilled if p is unramified in K/Q. Corollary 2.5. Assume p = 3. If 3 does not divide the order of Cl F ( ω 1 ) and it is unramified in K, then λ(k) = λ(f ) = 0. Proof. First of all, the Theorem applies for p = 3 also, since we still have (2.1) thanks to ζ( 1) = 1/12: moreover, K is clearly disjoint from Q( 3) = Q(ζ 3 ), as it is unramified, and F/K is ramified, so S =. But in this case we have ω 2 = 1, so the statement of the Theorem is that all Iwasawa invariants λ χ vanish for χ ˆ and their sum is precisely λ(k). Concerning λ(f ), in the proof of the Theorem we first prove that all λ χω vanish, and deduce from it the vanishing of their mirror parts. Remark 2.6. In the case K = Q( d) is real quadratic, this is a classical result of Scholtz (although it is expressed in term of Iwasawa invariants), see [Was] Theorem Theorem 2.7. Let K be a real quadratic field and suppose that 5 Cl F. Then λ(k) = 0. Proof. Write K = Q( d) and let χ be its non-trivial character: the result being well-known if d = 5 we assume throughout that d 5. Then we should consider two cases, namely 5 d and 5 d. We have the following 20

21 diagram of fields (we don t draw the whole of it): F = K(ζ 5 ) = K (ζ 5 ) G Q(ζ 5 ) H ϖ Q( 5) K = Q( d) K = Q( 5d) Q Suppose first of all that 5 d or that 5 is inert in K/Q. Since 5 [F : K], our hypothesis implies that 5 Cl K (see [Was] Lemma 16.15). But then we would trivially have λ(k) = 0 as an easy application of Nakayama s Lemma (see [Was] Proposition 13.22). We can thus suppose that 5 splits in K/Q. We then apply Theorem 2.3 to K instead of K: since Q( 5) Q(ζ 5 ), our field is linearly disjoint over Q from Q(ζ 5 ) and S = thanks to degree computations. Moreover the Main conjecture holds for F since F = K(ζ 5 ) and K is totally real and unramified at 5. We find that λ ω 2 χ = 0 where χ is the non-trivial character of K. But clearly χ = χω 2 so λ χ = 0. Since the Iwasawa invariant associated to the trivial character is λ(q) = 0 we have λ(k) = λ(q) + λ χ = 0. References [BN] R. Badino and T. Nguyen Quang Do, Sur les égalités du miroir et certaines formes faibles de la Conjecture de Greenberg, Manuscripta Mathematica, CXVI, (2005) [Gra] G. Gras, Class field theory: from theory to practice, SMM, Springer-Verlag 2005 [Gre] C. Greither, Class groups of abelian fields, and the main conjecture, Annales de l institut Fourier, XLII, (1992) [Was] L. Washington, Introduction to Cyclotomic Fields, GTM, Springer-Verlag

22 Luca Caputo Dipartimento di Matematica Università di Pisa Largo Bruno Pontecorvo, Pisa - ITALY caputo@mail.dm.unipi.it Filippo A. E. Nuccio Dipartimento di Matematica Università La Sapienza Piazzale Aldo Moro, Rome - ITALY nuccio@mat.uniroma1.it 22

23 Cyclotomic Units and Class Groups in Z p -extensions of Real Abelian Felds Filippo Alberto Edoardo Nuccio December 3 rd, 2008 Abstract For a real abelian number field F and for a prime p we study the relation between the p-parts of the class groups and of the quotients of global units modulo cyclotomic units along the cyclotomic Z p -extension of F. Assuming Greenberg s conjecture about the vanishing of the λ-invariant of the extension, a map between these groups has been constructed by several authors, and shown to be an isomorphism if p does not split in F. We focus in the split case, showing that there are, in general, non-trivial kernels and cokernels Mathematical Subject Classification: 11R23, 11R29 1 Introduction Let F/Q be a real abelian field of conductor f and let Cl F be its ideal class group. A beautiful formula for the order of this class group comes from the group of cyclotomic units: this is a subgroup of the global units O F whose index is linked to the order of Cl F. To be precise, we give the following definition ([Sin81], section 4): Definition 1.1. For integers n > 1 and a not divisible by n, let ζ n be a primitive n-th root of unity. Then Norm Q(ζn) F Q(ζ (1 n) ζa n) F and we define the cyclotomic numbers D F to be the subgroup of F generated by 1 and Norm Q(ζn) F Q(ζ (1 n) ζa n) for all n > 1 and all a not divisible by n. Then we define the cyclotomic units of F to be Cyc F := D F O F 23

24 Sinnott proved in [Sin81], Theorem 4.1 together with Proposition 5.1, the following theorem: Theorem (Sinnott). There exists an explicit constant κ F divisible only by 2 and by primes dividing [F : Q] such that [O F : Cyc F ] = κ F Cl F. Let now p be an odd prime that does not divide [F : Q]: by tensoring O F, Cyc F and Cl F with Z p we get an equality [O F Z p : Cyc F Z p ] = Cl F Z p and it is natural to ask for an algebraic interpretation of this. Moreover, observe that our assumption p [F : Q] makes the Galois group := Gal(F/Q) act on the modules appearing above through one-dimensional characters, and we can decompose them accordingly: in the sequel we write M(χ) for every Z[ ]-module M to mean the submodule of M Z p of M on which acts as χ, where χ ˆ (see the beginning of Section 3 for a precise discussion). Then an even more optimistic question is to hope for a character-by-character version of Sinnott s theorem, namely [O F Z p(χ) : Cyc F Z p (χ)]? = Cl F Z p (χ) (1.1) and then ask for an algebraic interpretation of this. Although it is easy to see that these -modules are in general not isomorphic (see the example on page 143 of [KS95]), it can be shown that they sit in an exact sequence for a wide class of fields arising in classical Iwasawa theory. More precisely, let F /F be the cyclotomic Z p -extension of F and let Γ = Gal(F /F ) = Z p : then F = n 0 F n... F n F n 1... F 0 = F where F n /F is a cyclic extension of degree p n whose Galois group is isomorphic to Γ/Γ pn. In a celebrated work (see [Iwa73]) Iwasawa gives a formula for the growth of the order of Cl Fn Z p : he proves that there are three integers µ, λ and ν, and an index n 0 0, such that Cl Fn Z p = p µpn +λn+ν for every n n 0. Moreover, Ferrero and Washington proved in [FW79] that the invariant µ vanishes. A long-standing conjecture by Greenberg (see [Gre76], where 24

25 conditions for this vanishing are studied) predicts that λ = 0: according to the conjecture the p-part of the class groups should stay bounded in the tower. Although a proof of this conjecture has never been provided, many computational checks have been performed verifying the conjecture in many cases (see, for instance, [KS95]). Under the assumptions λ = 0 and χ(p) 1, i. e. p does not split in F, some authors (see [BNQD01], [KS95], [Kuz96] and [Oza97]) were able to construct an explicit isomorphism α : ( Cl Fn Z p ) (χ) = ( O F n /Cyc Fn Z p ) (χ) (1.2) if n is big enough. Although the construction of the above morphism works also in the case χ(p) = 1, as detailed in the beginning of Section 5, the split case seems to have never been addressed. We focus then on this case, and study the map in this contest, still calling it α. Our main result is the following (see Corollary 5.2) Theorem. With notations as above, assume that χ is a character of such that χ(p) = 1 and that λ = 0. Then, for sufficiently big n, there is an exact sequence 0 K ( Cl Fn Z p ) (χ) α ( O F n /Cyc Fn Z p ) (χ) C 0 : both the kernel K and the cokernel C of α are cyclic groups with trivial Γ-action of order L p (1, χ) 1 p where L p (s, χ) is the Kubota-Leopoldt p-adic L-function. Acknowledgments This work is part of my PhD thesis, written under the supervision of René Schoof. I would like to take this opportunity to thank him not only for proposing me to work on this subject and for the help he gave me in writing this paper, but especially for all the time and patience he put in following me through my PhD and for the viewpoint on Mathematics he suggested me. 2 Some Tate Cohomology In this section we briefly recall some well-known facts that are useful in the sequel. Throughout, L/K is a cyclic extension of number fields, whose Galois group we denote by G. In our application, K and L will usually be layers F m and F n of the cyclotomic Z p -extension for some n m, but we prefer here not to restrict to this special case. 25